**Background:**

Let $u:\mathbb{R}^n\to \mathbb{R}$. Then the paper considers the following problem

\begin{align*} -\operatorname{div}(w(x) \nabla u) &= w(x) \text{ in } \Omega \\ u &= 0 \text{ on } \partial \Omega\\ u_\nu&= c\text{ on } \partial \Omega \\ \nabla^2\log w(\nabla u,\nabla u)+\frac{(\nabla \log w\cdot \nabla u)^2}{\alpha}&\leq 0 \text{ in }\Omega \end{align*} where $w$ is a homogenous weight such that $\nabla w \cdot x = \alpha x$ for $\alpha >0$ for eg. $w(x)=|x|^\alpha.$ Then the authors prove the following theorem:

Theorem 1.1Let $Ω$ be a connected and bounded domain with the smooth boundary $∂Ω$ in $\mathbb{R}^n$. If there exists a solution $u ∈ C^2(Ω)$ of the overdetermined problem above, then $Ω$ is a ball centered at 0.

Their strategy to prove this theorem relies on showing that $u$ is radial. However, their proof depends crucially on the inequality involving the weights given above. I am trying to verify if the inequality holds in general for $w(x)=|x|^\alpha.$ In the case when $u$ is assumed to be radial, the inequality is true. However, when proving **Theorem 1.1** one cannot assume that $u$ is radial (since that is the goal!) and as I have computed below, the inequality is false when $w(x)=|x|^\alpha.$

**Computation:**

First, $$\frac{(\nabla \log w\cdot \nabla u)^2}{\alpha}=\frac{\alpha}{|x|^4}(x\cdot \nabla u)^2$$

since

$$\nabla\log w = \frac{\alpha}{|x|^2} x.$$

I am guessing that (the authors did not define this in the paper)

$$\nabla^2\log w(\nabla u,\nabla u)=\sum_{i,j}D_{ij}\log wD_iuD_ju$$

where $D_i u$ is the partial derivative wrt to $x_i$ of $u.$ This gives me

$$\nabla^2\log w(\nabla u,\nabla u)=\frac{\alpha |\nabla u|^2}{|x|^2}-\frac{2\alpha (x\cdot \nabla u)^2}{|x|^4}$$

since

$$D_{ij}\log w = \frac{\alpha}{|x|^2}\delta_{ij}-\frac{2\alpha}{|x|^4}x_i x_j.$$

Therefore, $$\nabla^2\log w(\nabla u,\nabla u)+\frac{(\nabla \log w\cdot \nabla u)^2}{\alpha}=\frac{\alpha |\nabla u|^2}{|x|^2}-\frac{\alpha (x\cdot \nabla u)^2}{|x|^4}\\ =\frac{\alpha |\nabla u|^2}{|x|^2}\left(1-\cos^2(\theta_x)\right)\geq 0,$$

where $\theta_x$ is the angle between $x$ and the gradient of $u$, however, this contradicts the claim in the paper. Where have I made a mistake?